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Bihermitian Structures on Complex Surfaces
Author(s) -
Apostolov V.,
Gauduchon P.,
Grantcharov G.
Publication year - 1999
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/s0024611599012058
Subject(s) - mathematics , torus , pure mathematics , divisor (algebraic geometry) , surface (topology) , conformal map , dual (grammatical number) , complex torus , k3 surface , manifold (fluid mechanics) , geometry , combinatorics , mechanical engineering , art , literature , engineering , moduli space
Bihermitian complex surfaces are oriented conformal four‐manifolds admitting two independent compatible complex structures. Non‐anti‐self‐dual bihermitian structures on R 4 and the four‐dimensional torus T 4 have recently been discovered by P. Kobak. We show that an oriented compact 4‐manifold, admitting a non‐anti‐self‐dual bihermitian structure, is a torus or K3 surface in the strongly bihermitian case (when the two complex structures are independent at each point) or, otherwise, must be obtained from the complex projective plane or a minimal ruled surface of genus less than 2 by blowing up points along some anti‐canonical divisor (but the actual existence of bihermitian structures in the latter case is still an open question). The paper includes a general method for constructing non‐anti‐self‐dual bihermitian structures on tori, K3 surfaces and S 1 × S 3 . Further properties of compact bihermitian surfaces are also investigated. 1991 Mathematics Subject Classification : 53C12, 53C55, 32J15.